# Download Problems of Potential Theory by Evans G. C. PDF

By Evans G. C.

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1). We get Pkx\t) = j^{Pkxm n =PkC^\jPMt)+Pkm = Xk(Pkx)it) + iPkf)(t). 6 [80], since Pk € L{X). 1, P^x € AA{X). We conclude that n ^(*) = E^^^(*)^^^w A;=l as a finite sum of ahnost automorphic functions. D. 4. Let A be a linear operator C^ -> C^ and f{t) : R -> C^ an almost automorphic function. 1) is almost automorphic. An/ where A i , . . , A^ are the eigenvalues of A. 1) and put y{t) = B~^x(t). 6 [80]). ,gn{t)) = B-'f{t), teR. 1). We can say that yn-i(<) is also almost automorphic and proceed until y\{t), which proves that y G AA{X) and consequently x = By € AA{X) too.

Proof: Denote by C{R,E) the linear space of all continuous bounded functions R -^ E and by (9^), n € N, the family of seminorms which generates the topology r od E. Without loss of a generality we may assume that Qn-ti ^ Qm pointwise, for n € N. 36 1 Introduction and Preliminaries Define q^{f):=swpqn{f{x)), n € N. Obviotisly (q^) form a family of seminorms of C(M, E). Moreover, it is clear that q^_^i > q^ for n G N. Define the pseudo-norm ifi't^TTm "" ^^^"'•^'Obviously C{R^E) with the above defined pseudo-norm is a Prechet space.

Define qn := Pi Vp2 V . . Vpn for n G N. We have gn+i > Qn for neN. The space s considered with the family of seminorms (qn) is a Prechet space. 210) that each closed and bounded subset of s is compact. Thus, in particular, s is not a Banach space. 6 [80], s is perfect. 58 to functions of two variables of the form f{t,x) (see for instance [16]) as follows. 67. A continuous function f :Rx E -^ E is said to be almost periodic in t for each x E E, if for each neighbourhood of the origin U, there exists a real number I > 0, such that every compact interval of the real line contains at least a point r such that f(t + r, x) - / ( t , x) e U, for each t € M and x e E.